I took a look a photos of the regulator for the 1998 Ford Mustang. My guess is your noise is caused by something that is loose.
It appears that there is a torsion spring on the regulator that increases tension as the window lowers. This is probably there to help the motor push the window back up. Since your motor is making the popping noises when the window reaches the bottom of travel, my guess is that the spring tension is increasing the forces on the motor to the point where the motor gear is slipping out of the regulator gear. If you assembled the motor to the regulator with screws, then they may have worked loose, or if you used rivits, they might not have been tight enough in the first place.